Abelian
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In mathematics, the term abelian is used in many different definitions:
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In group theory
- An abelian group is a group in which the binary operation is commutative.
- The category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms.
- A metabelian group is a group where the commutator subgroup is contained in the center.
- Any group is "made abelian" by its abelianisation.
In Galois theory
- An abelian extension is a field extension for which the associated Galois group is abelian.
In real analysis
- Abelian theorems are used in the summation of divergent series.
In functional analysis
- An abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
In topology and number theory
- An abelian variety is a complex torus that can be embedded into projective space. This includes the elliptic curves and the abelian varieties of CM-type.
- An abelian function is a meromorphic function on an abelian variety.
- An abelian integral is a function related to the indefinite integral of a differential of the first kind. This includes the elliptic integrals.
In category theory
- A preabelian category is an additive category that has all kernels and cokernels.
- An abelian category is a preabelian category in which morphisms and objects can be added.